Page 142 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

132 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
We now use the convolution principle to evaluate the solution for the general case of
q(τ). We are searching for the inverse transform of
√ √
1 cosh(ς s ) Q(s ) cosh(ς s )
√ + 1 − √
s cosh s s cosh s
The inverse transform of the first term is given above. As for the second term, the inverse
transform of Q(s )issimply q(τ) and the inverse transform of the second term, absent Q(s) is

2(−1) n+1 cos 2n−1 ςπ 2n−1 2 2
1 − 2 e −( 2 ) π τ
(2n − 1)π
According to the convolution principle, and summing over all poles

∞
2(−1) n+1 cos 2n−1 ςπ 2n−1 2 2
u(ς, τ) = 2 e −( 2 ) π τ
(2n − 1)π
n=1

∞ τ n+1 2n−1
2(−1) cos 2 ςπ −( 2n−1 2 2

+ 1 − e 2 ) π τ q(τ − τ )dτ
τ =0 (2n − 1)π

n=1
Example 8.6. Next consider heat conduction in a semiinfinite region x > 0, t > 0. The initial
temperature is zero and the wall is subjected to a temperature u(0, t) = f (t)atthe x = 0
surface.
u t = u xx
u(x, 0) = 0
u(0, t) = f (t)

and u is bounded.
Taking the Laplace transform and applying the initial condition

sU = U xx

Thus
√ √
U(x, s ) = A sinh x s + B cosh x s
Both functions are unbounded for x →∞. Thus it is more convenient to use the
equivalent solution
√ √ √
U(x, s ) = Ae −x s + Be x s = Ae −x s

in order for the function to be bounded. Applying the boundary condition at x = 0

F(s ) = A